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%I #16 Jul 09 2024 08:55:46
%S 2,6,28,1171,337664,477339616,2872202032640,72057595967392816,
%T 7462505059899322983424,3169126500571074529242309120,
%U 5492677668532710795071526353530880,38716571525226776289479030777920527620096,1106936151351216411420552029913564178922327982080
%N Number of ways of tiling the n X n torus up to 90-degree rotations of the square by two tiles that are each fixed under 90-degree rotation of the square.
%H S. N. Ethier and Jiyeon Lee, <a href="http://arxiv.org/abs/1502.03792">Counting toroidal binary arrays, II</a>, arXiv:1502.03792v1 [math.CO], Feb 12, 2015 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Lee/lee6.html">J. Int. Seq. 18 (2015)</a>.
%H Peter Kagey, <a href="/A368143/a368143.pdf">Illustration of a(3)=28</a>
%H Peter Kagey and William Keehn, <a href="https://arxiv.org/abs/2311.13072">Counting tilings of the n X m grid, cylinder, and torus</a>, arXiv: 2311.13072 [math.CO], 2023. See also <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Kagey/kagey6.html">J. Int. Seq.</a>, (2024) Vol. 27, Art. No. 24.6.1, pp. A-21, A-24.
%t A368143[n_] := 1/(4n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 2^(n^2/LCM[c, d])]]]] + n^2*If[OddQ[n], 2^((n^2 + 1)/2) + 2^((n^2 + 7)/4), 7*2^((n^2 - 4)/2) + 5*2^(n^2/4)])
%Y Cf. A047937, A367534, A368144, A368145.
%K nonn
%O 1,1
%A _Peter Kagey_, Dec 16 2023